Department of

# Mathematics

Seminar Calendar
for number theory events the next 6 month of Friday, July 1, 2016.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, July 5, 2016

11:00 am in 343 Altgeld Hall,Tuesday, July 5, 2016

#### Asymptotic distribution of Beurling's generalized prime numbers

###### Jasson Vindas (Ghent University, Belgium)

Abstract: This talk is a survey of classical and recent results on the asymptotic distribution of generalized prime numbers. The goal of Beurling's generalized prime number theory is to replace the ordinary prime numbers by a rather arbitrary non-decreasing sequence of positive real numbers (generalized primes), consider then the multiplicative semigroup generated by it (generalized integers), and then establish relations between asymptotic properties of the counting functions of the generalized primes and integers. We will discuss various conditions that ensure the validity of the prime number theorem in this context.

2:00 pm in 343 Altgeld Hall,Tuesday, July 5, 2016

#### Exact Wiener-Ikehara theorems

###### Wen-Bin Zhang (University of the West Indies, Kingston, Jamaica and UIUC)

Abstract: The famous tauberian theorem of Wiener and Ikehara gives a sufficient condition for deducing the asymptotic behavior of a function from information about its Laplace transform. In this talk, we show that the "nondecreasing" condition can be replaced by a weaker one, and this set of conditions is necessary as well as sufficient.

Thursday, July 7, 2016

11:00 am in 343 Altgeld Hall,Thursday, July 7, 2016

#### PNT equivalences for Beurling numbers

###### Gregory Debruyne (Ghent University, Belgium)

Abstract: In classical prime number theory several asymptotic relations are considered to be "equivalent" with the prime number theorem, meaning that they could be deduced by simple real variable arguments. In the context of Beurling numbers, however, this is no longer the case: sometimes extra hypotheses have to be imposed to show the equivalence between different asymptotic relations. We will present some recent results on the subject. In contrast to the earlier work of Diamond and Zhang, who used strictly elementary methods, our approach will make extensive use of the zeta function of the prime number system and can thus no longer be regarded as elementary. The talk is based on collaborative work with Jasson Vindas.

Thursday, August 25, 2016

11:00 am in 241 Altgeld Hall,Thursday, August 25, 2016

#### Organizational Meeting

Thursday, September 1, 2016

11:00 am in 241 Altgeld Hall,Thursday, September 1, 2016

#### A lower bound for the least prime in an arithmetic progression

###### Junxian Li (Illinois Math)

Abstract: Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek lower bounds for $P(k)$. In particular, we show that for almost every $k$ one has $P(k) \gg \phi(k) \log k \log_2 k \log_4 k / \log_3 k,$ answering a question of Ford, Green, Konyangin, Maynard, and Tao. We rely on their recent work on large gaps between primes. Our main new idea is to use sieve weights to capture not only primes, but also small multiples of primes. We also give a heuristic which suggests that $\liminf_{k} \frac{P(k)}{ \phi(k) \log^2 k} = 1$. This is joint work with Kyle Pratt and George Shakan.

Thursday, September 8, 2016

11:00 am in 241 Altgeld Hall,Thursday, September 8, 2016

#### Galois actions on the homology of Fermat curves, and applications

###### Vesna Stojanoska (Illinois Math)

Abstract: In the late 80ties, Anderson gave a method for describing the Galois action on the singular homology of Fermat curves. In this talk, I will concentrate on Fermat curves of prime exponent p, and the singular homology will have mod p coefficients. I will describe how to explicitly determine the Galois action using Anderson’s work, and then proceed to compute some Galois cohomology groups, which naturally appear when studying certain obstructions to the existence of rational points. This is all joint work in progress with R. Davis, R. Pries, and K. Wickelgren.

Tuesday, September 13, 2016

2:00 pm in 241 Altgeld Hall ,Tuesday, September 13, 2016

#### Sieve methods and Fouvry-Iwaniec primes

###### Kyle Pratt   [email] (UIUC )

Abstract: Sieve methods were invented in order to find prime numbers in various sequences. In their original incarnation sieves are unfortunately incapable of fulfilling their intended purpose, due to a fundamental obstruction known as the "parity problem''. However, with additional analytic input it is sometimes possible to break the parity barrier and find primes in interesting sequences. In this talk, the first in a series of two lectures, we begin exploring the result of Fouvry and Iwaniec that there are infinitely many primes $p$ of the form $p = x^2+\ell^2$, where $\ell$ itself is a prime. We discuss the "sieve-theoretic'' aspects of the proof, and study an interesting spacing problem for rational points coming from roots of a quadratic congruence.

Thursday, September 15, 2016

11:00 am in 241 Altgeld Hall,Thursday, September 15, 2016

#### OverGuassian Polynomials

###### Byungchan Kim (Seoul National University of Science and Technology)

Abstract: We introduce a two variable generalization of Gaussian polynomial. After introducing some basic properties analogous to classical binomial coefficients, I will discuss its roles in q-series and integer partitions. This is joint work with Jehanne Dousse.

Tuesday, September 20, 2016

2:00 pm in 241 Altgeld Hall ,Tuesday, September 20, 2016

#### Sieve methods and Fouvry-Iwaniec primes

###### Kyle Pratt   [email] (UIUC )

Abstract: Sieve methods were invented in order to find prime numbers in various sequences. In their original incarnation sieves are, unfortunately, incapable of fulfilling their intended purpose, due to a fundamental obstruction known as the "parity problem.'' However, with additional analytic input it is sometimes possible to break the parity barrier and find primes in interesting sequences. In this talk, the second of two lectures, we continue our exploration of the result of Fouvry and Iwaniec that there are infinitely many primes of the form $p = x^2+\ell^2$, with $\ell$ a prime. We focus on the parity-breaking features of the proof, which requires finding cancellation in a bilinear form involving the $M\ddot{o}bius$ function.

Thursday, September 22, 2016

11:00 am in 241 Altgeld Hall,Thursday, September 22, 2016

#### Noncongruence modular form and Scholl representation

###### Tong Liu (Purdue University)

Abstract: In this talk, I will report the progress of the research on noncongruence modular form via the attached Galois representation (Scholl representation). As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. I will show this is the case for certain situations (potentially GL(2)-type) and explain how (potential) automorphy of Scholl representation relates to some standard conjectures of noncongruence modular form. This is joint work of Winnie Li and Ling Long.

Tuesday, September 27, 2016

2:00 pm in 241 Altgeld Hall,Tuesday, September 27, 2016

#### Distribution of the periodic points of the Farey map

###### Byron Heersink   [email] (UIUC )

Abstract: A result of Series established a cross section of the geodesic flow in the tangent space of the modular surface which provided a lucid explanation of the connection between the geodesics in the modular surface and continued fractions. Pollicott later utilized this connection to show the limiting distribution of the periodic points of the Gauss map, i.e., the periodic continued fractions, when ordered according to the length of corresponding closed geodesics. In this talk, we outline how to extend the work of Series and Pollicott to obtain results for the Farey map. In particular, we expand the cross section of Series so that the return map under the geodesic flow is a double cover of the Farey map's natural extension. We then show how to adapt the method of Pollicott, which uses the analysis of a certain nuclear operator on the disk algebra, to prove an equidistribution result for the periodic points of the Farey map.

Thursday, September 29, 2016

11:00 am in 241 Altgeld Hall,Thursday, September 29, 2016

#### Ramanujan's formula for $\zeta(2n+1)$

###### Bruce Berndt (Illinois Math)

Abstract: Let $\zeta(s)$ denote the Riemann zeta function. If $n$ is a positive integer, a famous formula of Euler provides an elegant evaluation of $\zeta(2n)$. However, little is known about $\zeta(2n+1)$. In Ramanujan's earlier notebooks, we find a formula for $\zeta(2n+1)$ which is a natural analogue of Euler's formula. We provide its history, indicate why it is "interesting," and show its connections with other mathematical objects such as the Dedekind eta function, Eisenstein series, and period polynomials.

Tuesday, October 4, 2016

2:00 pm in 241 Altgeld Hall,Tuesday, October 4, 2016

###### George Shakan (UIUC )

Abstract: Recently, Bourgain, Demeter and Guth showed that Vinogradov's mean value theorem deserves its name of "theorem." The conjecture followed from their general decoupling inequality. I'll consider the 2 dimensional case of their proof and explain some of the key inputs. In this case, their theorem asserts that the $L^6$ norm of the exponential sum $\sum_{j < N} e(j x + j^2 y)$ is $\ll N^{1/2 + \epsilon}$. Some of the key inputs are the translation invariance (in j), induction on scales, and a Kakeya-type estimate. This last estimate eventually follows from the elementary geometric fact that the intersection of two same-sized parallelograms of significantly different slopes have small intersection.

Thursday, October 6, 2016

11:00 am in 241 Altgeld Hall,Thursday, October 6, 2016

#### A proof of $M(x) = o(x)$ for Beurling generalized numbers

###### Harold Diamond (Illinois Math)

Abstract: In classical prime number theory there are several asymptotic formulas said to be equivalent'' to the Prime Number Theorem. One of these assertions is that $M(x)$, the summatory function of the Moebius function, is $o(x)$. Implications between these formulas are different for Beurling generalized numbers ($g$-numbers). We deduce the $g$-number version of $M(x) = o(x)$ using the PNT and a crude $O$-bound on the distribution of $g$-integers.

Tuesday, October 11, 2016

2:00 pm in 241 Altgeld Hall,Tuesday, October 11, 2016

#### The functional equation of the Dedekind zeta function of a number field

###### Ravi Donepudi (UIUC )

Abstract: Riemann’s seminal paper of 1859 establishes (among other things) the analytic continuation and functional equation of the Riemann zeta function. These results have since been generalized to several other similarly defined functions. Indeed, Wikipedia lists at least 33 "zeta functions" defined in diverse areas of mathematics. In this talk, we will discuss one such class of objects: the Dedekind zeta function attached to a number field. Erich Hecke was the first to prove a functional equation for these functions. We will describe Hecke’s proof in detail, highlighting it’s similarities to Riemann’s original proof. As a corollary we will be able to better see why important invariants of a number field, like the discriminant and regulator, mysteriously appear in the class number formula. People of both algebraic and analytic persuasions should find something interesting in this talk. If I use fancy words like Adeles or Ideles in the talk, you can shoot me.

Thursday, October 13, 2016

11:00 am in 241 Altgeld Hall,Thursday, October 13, 2016

#### Analyzing rationals by simpler rationals

###### Joseph Vandehey (Ohio State)

Abstract: The (decimal) expansions of rational numbers continue to be a rather mysterious object. For instance, we expect most rational numbers to have a periodic expansion that involves a 7, but how soon should we expect that 7 to appear? In this talk, we will discuss a new method of relating the expansion of a given rational number to rationals with smaller denominators, by means of a new differencing method for exponential sums that is highly effective for exponential sums with an exponential argument.

Tuesday, October 18, 2016

2:00 pm in 241 Altgeld Hall,Tuesday, October 18, 2016

#### Higher order mollifications of the zeta function

###### Nicolas Robles (UIUC )

Abstract: We will discuss the process of mollifying the Riemann zeta-function and how this can be used to obtain mean value moments and proportions of zeros. We will also describe how to mollify $1/(\zeta + \zeta' + \zeta'' +...)$ and discuss some properties of the associated exponential sums.

Thursday, October 20, 2016

11:00 am in 241 Altgeld Hall,Thursday, October 20, 2016

#### Finding integers from orbits of thin subgroups of SL(2, Z)

###### Xin Zhang (Illinois Math)

Abstract: Let $\Lambda < SL(2,\mathbb{Z})$ be a finitely generated, non-elementary Fuchsian group of the second kind, and $v,w$ be two primitive vectors in $\mathbb{Z}^2-0$. We consider the set $\mathcal{S}=\{\langle v\gamma,w\rangle_{\mathbb{R}^2}:\gamma\in\Lambda\}$, where $\langle\cdot,\cdot\rangle_{\mathbb{R}^2}$ is the standard inner product in $\mathbb{R}^2$. Using Hardy-Littlewood circle method and some infinite co volume lattice point counting techniques developed by Bourgain, Kontorovich and Sarnak, together with Gamburd's 5/6 spectral gap, we show that if $\Lambda$ has parabolic elements, and the critical exponent $\delta$ of $\Lambda$ exceeds 0.995371, then a density-one subset of all admissible integers (i.e. integers passing all local obstructions) are actually in $\mathcal{S}$, with a power savings on the size of the exceptional set (i.e. the set of admissible integers failing to appear in $\mathcal{S}$). This supplements a result of Bourgain-Kontorovich, which proves a density-one statement for the case when $\Lambda$ is free, finitely generated, has no parabolics and has critical exponent $\delta>0.999950$.

Tuesday, October 25, 2016

2:00 pm in 241 Altgeld Hall,Tuesday, October 25, 2016

#### Some results related to the distribution of zeros of a family of Dirichlet series.

###### Paulina Koutsaki (UIUC )

Abstract: Let $G(s)=\sum\limits_{n=1}^{\infty} a_n \,n^{-s}$ be a Dirichlet series with coefficients bounded by $n^{\epsilon}$ for every $\epsilon>0$, and define $\beta_{k,G}$ to be the supremum of the real parts of zeros of combinations of $G$ and its $k$ first derivatives. In this talk, we give an asymptotic formula for the number $\beta_{k,G}$ and investigate in more detail the case of Dirichlet L-functions. We will also discuss an inverse-type problem for the Riemann-zeta function. In particular, we compute the degree of the largest derivative needed for such a combination to vanish at a given real number. For example, a combination that vanishes at $\beta=1,000,000$ will involve a derivative of order at least 2,178,301. This is joint work with A. Tamazyan and A. Zaharescu.

Thursday, October 27, 2016

11:00 am in 241 Altgeld Hall,Thursday, October 27, 2016

#### Singular overpartitions

###### Ae Ja Yee (Penn State)

Abstract: Singular overpartitions, which were defined by George Andrews, are overpartitions whose Frobenius symbols have at most one overlined entry in each row. In his paper, Andrews obtained interesting results on singular overpartitions; in particular, one result relates a certain type of singular overpartitions with a subclass of overpartitions. In this talk, I will introduce partitions with dotted parity blocks and give a combinatorial proof of Andrews' result. I will also discuss some refinements on Andrews' result.

Tuesday, November 1, 2016

2:00 pm in 241 Altgeld Hall,Tuesday, November 1, 2016

#### An adaptation of the Langlands Correspondence in the contexts of Geometry and Quantum Physics

###### Georgios Kydonakis (UIUC )

Abstract: The Langlands Program emerged in the late 1960s as a web of conjectures relating deep questions in number theory, algebraic geometry and the theory of automorphic forms. While in the case of number fields the Langlands Correspondence is still a conjecture (except for special cases), in the case of function fields it is a theorem. A geometric reformulation of the correspondence was suggested in the 1980s, which would make sense over algebraic curves defined over $\mathbb{C}$, i.e. Riemann Surfaces. More recently, a relation between Langlands duality in Mathematics and S-duality in Quantum Field Theory has been established. We will discuss elements of these adaptations through the prism of "analogy" in Mathematics, the way A. Weil described it in his famous 1940 letter from jail.

Tuesday, November 8, 2016

2:00 pm in 241 Altgeld Hall,Tuesday, November 8, 2016

#### Langlands Correspondence and L-functions

###### Hao Sun (UIUC )

Abstract: I will talk about the langlands correspondence between Artin L-functions and the L-functions of automorphic representations.

Thursday, November 10, 2016

11:00 am in 241 Altgeld Hall,Thursday, November 10, 2016

#### A new approach to Waldspurger's formula

###### Rahul Krishna (Northwestern University)

Abstract: I will present a new trace formula approach to Waldspurger's formula for toric periods of automorphic forms on $PGL_2$. The method is motivated by interpreting Waldspurger's result as a period relation on $SO_2 \times SO_3$, which leads to a strange comparison of relative trace formulas. I will explain the local results needed to carry out this comparison, mention some technical hurdles, and discuss some optimistic dreams for extending these results to high rank orthogonal groups.

Tuesday, November 15, 2016

2:00 pm in 241 Altgeld Hall,Tuesday, November 15, 2016

#### Bounds on multiplicative character sums over a finite field

###### Chieu Minh Tran (UIUC )

Abstract: We will discuss a folkloric result on weak bounds for multiplicative character sums whose summation domain is a non-smooth variety over a finite field. We will briefly talk about a proof based on ell-adic cohomology, the Grothendieck-Lefschetz trace formula and Deligne's theorem in Weil II. Then, we will talk about a more elementary proof using model-theoretic techniques and a Weil-style bound for multiplicative character sums on a curve. If time permits, we will also explain the model-theoretic context where we need this result.

Thursday, November 17, 2016

11:00 am in 241 Altgeld Hall,Thursday, November 17, 2016

#### Consecutive primes and Beatty sequences

###### William Banks (University of Missouri)

Abstract: Beatty sequences are generalized arithmetic progressions which have been studied intensively in recent years. Thanks to the work of Vinogradov, it is known that every Beatty sequence S contains "appropriately many" prime numbers. For a given pair of Beatty sequences S and T, it is natural to wonder whether there are "appropriately many" primes in S for which the next larger prime lies in T. In this talk, I will show that this is indeed the case if one assumes a certain strong form of the Hardy-Littlewood conjectures. This is recent joint work with Victor Guo.

Wednesday, November 30, 2016

4:00 pm in 245 Altgeld Hall,Wednesday, November 30, 2016

#### Distribution of Sequences In Number Theory

###### Sneha Chaubey (UIUC Math)

Abstract: Given a sequence of real numbers, it is natural to ask how is the sequence distributed. For example, are there sub intervals that contain the same number of elements? What is the spacing between any two elements of the sequence? What is the probability distribution of the spacings between neighbors and so on. In this talk, we will consider some of these questions and will discuss techniques used to understand the finer structure of sequences. The talk will be accessible to all and easy to follow.

Thursday, December 1, 2016

11:00 am in 241 Altgeld Hall,Thursday, December 1, 2016

#### Chebyshev's bias for products of $k$ primes

###### Xianchang Meng (Illinois Math)

Abstract: For any $k\geq 1$, we study the distribution of the difference between the number of integers $n\leq x$ with $\omega(n)=k$ or $\Omega(n)=k$ in two different arithmetic progressions, where $\omega(n)$ is the number of distinct prime factors of $n$ and $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity . Under some reasonable assumptions, we show that, if $k$ is odd, the integers with $\Omega(n)=k$ have preference for quadratic non-residue classes; and if $k$ is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Hudson. However, the integers with $\omega(n)=k$ always have preference for quadratic residue classes. Moreover, as $k$ increases, the biases become smaller and smaller for both of the two cases.

Tuesday, December 6, 2016

2:00 pm in 241 Altgeld Hall,Tuesday, December 6, 2016

#### The eta function

###### Scott Ahlgren   [email] (UIUC)

Abstract: The Dedekind eta function is one of the fundamental functions of number theory. It is a basic building block for many types of modular forms, and it plays a central role in a range of number-theoretic problems. I will give a bunch of examples of objects which can be built using the eta function, and of the applications which are related to its properties.

2:00 pm in 241 Altgeld Hall,Tuesday, December 6, 2016

#### The eta function

###### Scott Ahlgren   [email] (UIUC)

Abstract: The Dedekind eta function is one of the fundamental functions of number theory. It is a basic building block for many types of modular forms, and it plays a central role in a range of number-theoretic problems. I will give a bunch of examples of objects which can be built using the eta function, and of the applications which are related to its properties.